A Further Analysis of the Convergence Rates and Patterns of the
Binomial Models
San-Lin Chung
Department of Finance, National Taiwan University
No. 85, Section 4, Roosevelt Road,
Taipei 106, Taiwan, R.O.C.
Tel: 886-2-33661084
Email: chungs@management.ntu.edu.tw
Pai-Ta Shih
Department of Economics, National Dong Hwa University
No. 1, Section 2, Da Hsueh Road,
Shoufeng, Hualien 974, Taiwan, R.O.C.
Tel: 886- 3-8635541
Email: bdshih@mail.ndhu.edu.tw
Abstract
This paper extends the generalized Cox-Ross-Rubinstein (hereafter GCRR) model
of Chung and Shih (2007). We provide a further analysis of the convergence rates and
patterns based on various GCRR models. The numerical results indicate that the GCRRXPC model and the GCRR-JR ( p  1 2 ) model (defined in Table 1) outperform the other
GCRR models for pricing European calls and American puts. Our results confirm that the
node positioning and the selection of the tree structures (mainly the asymptotic behavior
of the risk-neutral probability) are important factors for determining the convergence
rates and patterns of the binomial models.
JEL Subject Classification: G13.
Key words: binomial model, rate of convergence, monotonic convergence
1
1. Brief Review of the Binomial Models
Ever since the seminal works of Cox, Ross, and Rubinstein (1979, hereafter CRR)
and Rendleman and Bartter (1979), many articles have extended the binomial option
pricing models in many aspects. One stream of the literature modifies the lattice or tree
type to improve the accuracy and computational efficiency of binomial option prices. The
pricing errors in the discrete-time models are mainly due to “distribution error” and
“nonlinearity error” (see Figlewski and Gao (1999) for thorough discussions). Within the
literature, there are many proposed solutions that reduce the distribution error and/or
nonlinearity error.
It is generally difficult to reduce the distribution error embedded in the binomial
model when the number of time steps ( n ) is small, probably due to the nature of the
problem. Nevertheless there are two important works which might shed some light on
understanding or dealing with distribution error. First, Omberg (1988) developed a family
of efficient multinomial models by applying the Gauss-Hermite quadrature 1 to the
integration problem (e.g. N (d1 ) in the Black-Scholes formula) presented in the option
pricing formulae. Second, Leisen and Reimer (1996) modified the sizes of up- and downmovements by applying various normal approximations (e.g. the Camp-Paulson inversion
formula) to the binomial distribution derived in the mathematical literature.
Concerning the techniques to reduce nonlinearity error, there are least three
important works (methods) worth noting in the literature. First, Ritchken (1995) and Tian
(1999) showed that one can improve the numerical accuracy of the binomial option prices
by allocating one of the final nodes on the strike price or one layer of the nodes on the
barrier price. Second, Figlewski and Gao (1999) proposed the so-called adaptive mesh
method which sharply reduces nonlinearity error by adding one or more small sections of
fine high-resolution lattice onto a tree with coarser time and price steps. Third, Broadie
and Detemple (1996) and Heston and Zhou (2000) suggested modified binomial models
by replacing the binomial prices prior to the end of the tree by the Black-Scholes values,
or by smoothing payoff functions at maturity, and computing the rest of binomial prices
as usual.
1
The Gauss-Hermite quadrature is a well developed and highly efficient method in the numerical
integration literature.
2
Another stream of research focuses on the convergence rates and patterns of the
binomial models. For example, Leisen and Reimer (1996) proved that the convergence is
of order O (1 / n) for the CRR model, the Jarrow and Rudd (1983) model, and the Tian
(1993) model. Heston and Zhou (2000) also showed that the rate of convergence for the
lattice approach can be enhanced by smoothing the option payoff functions and cannot
exceed 1 / n at some nodes of the tree. Concerning the convergence patterns of the
binomial models, it is widely documented that the CRR binomial prices converge to the
Black-Scholes price in a wavy, erratic way (for example, see Broadie and Detemple
(1996) and Tian (1999)).2 Several remedies have been proposed in the literature, e.g. see
Broadie and Detemple (1996), Leisen and Reimer (1996), Tian (1999), Heston and Zhou
(2000), Widdicks et al. (2002), and Chung and Shih (2007).
In a recent article, Chung and Shih (2007) proposed a generalized CRR (hereafter
GCRR) model by adding a stretch parameter into the CRR model. With the flexibility
from the stretch parameter, the GCRR model provides a numerically efficient method for
pricing a range of options. Moreover, Chung and Shih (2007) also showed that (1) the
convergence pattern of the binomial prices to the accurate price depends on the node
positioning (see their Theorem 2) and (2) the rate of convergence depends on the
selection of the tree structures (i.e. the risk-neutral probability p and jump sizes u and d).
In this paper, we provide a further analysis of the convergence rates and patterns based on
the GCRR models. Our results confirm that the node positioning and the selection of the
tree structures are important factors for determining the convergence rates and patterns of
the binomial models.
The rest of the paper is organized as follows. Section 2 discusses the importance of
node positioning for monotonic convergence of binomial prices. We also summarize a
complete list of binomial models with monotonic convergence in Section 2. Section 3
introduces the GCRR model and discusses its flexibility for node positioning. Various
GCRR related models are also defined in Section 3. Section 4 presents the numerical
results of various GCRR models (such as the GCRR-XPC and GCRR-JR models) against
other binomial models for pricing a range of options. The numerical analysis is mainly
2
Please also see Diener and Diener (2004) for a complete discussion of the price oscillations in a tree
model.
3
focused the convergence rates and patterns of various binomial models. Section 5
concludes the paper.
2. The Importance of Node Positioning for Monotonic Convergence
Following the standard option pricing model, we assume that the underlying asset
price follows a geometric Brownian motion, i.e.
dS / S   dt   dW ,
(1)
where  is the drift rate,  is the instantaneous volatility, and W is a Wiener process.
Standard dynamic hedge and complete market argument assures that options can be
valued by assuming risk neutrality, i.e.   r . For pricing plain vanilla European options,
Black-Scholes had derived closed-form solutions. For pricing other types of options such
as American options, we have to rely on numerical approximation methods, e.g. a
binomial model.
In an n-period binomial model, the time to maturity of the option ( T ) is divided
into n equal time steps, i.e. t  T / n . If the current stock price is S, then the next period
price jumps either upward to uS with probability q or downward to dS with probability
1  q, where 0  d  e rt  u and 0  q  1 . Again the standard dynamic hedge approach
ensures that the risk-neutral probability p, instead of q, is relevant for pricing options.
The binomial option pricing model is completely determined by the jump sizes u and d
and the risk-neutral probability p. In the CRR model, three conditions, i.e. the risk-neutral
mean and variance of the stock price in the next period and ud  1 , are utilized to
determine u, d, and p as follows: u  e
t
, d  e
t
, and p  (ert  d ) /(u  d ).
It is widely documented that the CRR binomial prices converge to the BlackScholes price in a wavy, erratic way (for example, see Broadie and Detemple (1996) and
Tian (1999)).3 This kind of convergent pattern can be explained by the following theorem.
Theorem 1.
Let Cn and C BS be the n-period CRR binomial and continuous-time prices
of a standard European call option with the terminal payoff max( ST  X ,0) . Therefore,
3
Please also see Diener and Diener (2004) for a complete discussion of the price oscillations in a tree
model.
4
Cn  CBS 
 (0.5   (n)) 2
n
where   2 Xe  rT  (d 2 )   T , d 2 
 O(1/ n)
(2)
ln( S / X )  (r   2 / 2)T
ln( S x / X )
,  ( n) 
,
ln( u / d )
 T
S x  Su m d n m , and m is the smallest integer which satisfies Su m d nm  X .
Proof: Please see Chung and Shih (2007).
Therefore, Theorem 1 shows that the node positioning (i.e. the relative distance of
the strike price between the most adjacent nodes) is the main reason why the CRR
binomial prices converge to the Black-Scholes price in a wavy, erratic way. To get a
monotonic convergent pattern, we can (1) use the CRR model corrected with the
quadratic function (see equation (1)) as suggested by Chung and Shih (2007); (2) fine
tune the binomial model such that  (n) is a constant (see Tian (1999) ); (3) choose the
appropriate number of time steps n such that  (n) is a constant (see WAND (2002)); (4)
smooth the option payoff function to be a continuous and differentiable function (see
Heston and Zhou (2000)); (5) replace the binomial prices prior to the end of the tree by
the Black-Scholes values and computing the rest of binomial prices as usual (see Broadie
and Detemple (1996)); or (6) place the strike price in the center of the final nodes (see
Leisen and Reimer (1996) and Chung and Shih (2007)).
Once the monotonic convergent pattern is derived, various extrapolation techniques
are applicable to the above binomial models and the option price can be more efficiently
calculated. The extrapolation formula used in this article is briefly summarized in the
Appendix.
3. The Flexibility of GCRR Model for Node Positioning
In the GCRR model of Chung and Shih (2007), a stretch parameter (  ) is
incorporated into the CRR model so that the shape (or spanning) of the lattice can be
flexibly adjusted. In other words, the condition that ud  1 , which is used in the
traditional CRR model, is relaxed in the GCRR model. As a result, the GCRR model has
one more degree of freedom to adjust the shape of the binomial lattice. We restate the
GCRR model of Chung and Shih (2007) in Theorem 2.
5
Theorem 2.
In the GCRR model the jump sizes and probability of going up are as
follows:
u  e 
t
, d e
1
  t

, and p 
e rt  d
,
ud
(3)
where   R  is a stretch parameter which determines the shape (or spanning) of the
binomial tree. Moreover, when t  0 , i.e., the number of time steps n grows to infinity,
the GCRR binomial option prices converge to the Black-Scholes formulae for plain
vanilla European options.
The stretch parameter (  ) in the GCRR model gives us the flexibility for node
positioning. For instance, Chung and Shih (2007) placed the strike price in the center of
the final nodes (termed GCRR-XPC model in their paper) and showed that the GCRRXPC binomial prices converges to the accurate price smoothly and monotonically. A
straightforward extension of Chung and Shih (2007) is to place the strike price on a
certain place of binomial tree, e.g. on the 1/3-th or the 2/3-th nodes.
Morevoer, the idea of the GCRR model potentially can be applied to other binomial
models. For example, we can generalize the Jarrow and Rudd (1983, hereafter JR) model
by setting
1
( r   2 ) t  
1
p 2
, ue 2
 1
t
, and d  e
1
1
( r   2 ) t   t
2

.
(4)
By changing the stretch parameter  , the shape (or spanning) of the lattice can be
flexibly adjusted so that one of the final nodes can be allocated on the strike price.
In the following section, we will investigate the numerical performances of various
GCRR models where one of the final nodes can be allocated on the strike price and
therefore the convergent patterns are monotonic. In addition to CRR and JR models, six
GCRR related models are investigated in the next section and their definitions are given
in Table 1.
Table 1 The Definitions of Various GCRR Models
Model
GCRR-XPC
Definition
a GCRR model where the stretch parameter  is chosen such that the
strike price is placed in the center of final nodes
6
GCRR- X on the
1/3-th node
GCRR- X on the
2/3-th node
GCRR-JR (p=1/2)
GCRR-JR (p=1/3)
GCRR-JR (p=2/3)
a GCRR model where the stretch parameter  is chosen such that the
number of nodes above the strike price is one third of the total final nodes
and one of the final nodes can be allocated on the strike price
a GCRR model where the stretch parameter  is chosen such that the
number of nodes above the strike price is two third of the total final nodes
and one of the final nodes can be allocated on the strike price
a GCRR-JR model where the stretch parameter  is chosen such that
the up probability is the most close to 1/2 and one of the final nodes can be
allocated on the strike price
a GCRR-JR model where the stretch parameter  is chosen such that
the up probability is the most close to 1/3 and one of the final nodes can
be allocated on the strike price
a GCRR-JR model where the stretch parameter  is chosen such that
the up probability is the most close to 2/3 and one of the final nodes can
be allocated on the strike price
4. Numerical Results of Various GCRR Models
In this section, we will investigate the convergence pattern, the rate of convergence,
and the accuracy of six GCRR models defined in Table 1 against the CRR model and the
JR model. Figure 1 shows the convergence pattern of the CRR model, the GCRR-XPC
model, GCRR- X on the 1/3-th node model, and GCRR- X on the 2/3-th node model for
pricing an in-the-money call option. The parameters in Figure 1 are adopted from Tian
(1999): the asset price is 100, the strike price is 95, the volatility is 0.2, the maturity of
the option is six months, and the risk-free rate is 0.06. The pricing error is defined as the
difference between the binomial price and the Black-Scholes price. In Figure 1, the
pricing errors are plotted against an even number of time steps ranging from 10 to 100.
It is apparent from Figure 1 that the convergence of the CRR model is not
monotonic and smooth. Moreover, the pricing errors oscillate between positive and
negative values. In contrast, the GCRR-XPC model, GCRR- X on the 1/3-th nodes
model, and GCRR- X on the 2/3-th nodes model converge monotonically to the BlackScholes price. Therefore, various extrapolation techniques are applicable to the GCRRXPC model, GCRR- X on the 1/3-th node model, and GCRR- X on the 2/3-th node
model. A similar convergence pattern is also embedded in GCRR-JR (p=1/2) model,
GCRR-JR (p=1/3) model and GCRR-JR (p=2/3) model as shown in Figure 2.
7
Figure 1 The Convergence Patterns of the CRR Model and Three Related GCRR models
0.15
CRR
GCRR-XPC
GCRR- X on the 1/3-th nodes
0.1
GCRR- X on the 2/3-th node
Pricing Error
0.05
0
-0.05
-0.1
-0.15
-0.2
10
28
46
64
82
100
Number of Time Steps
This figure shows the convergence patterns of the CRR model, GCRR-XPC model, GCRR- X on the 1/3-th
node model, and GCRR- X on the 2/3-th node model. The asset price is 100, the strike price is 95, the
volatility is 0.2, the maturity of the option is six months, and the risk-free rate is 0.06.
Figure 2 The Convergence Patterns of the JR Model and Three Related GCRR models
0.05
JR
GCRR-JR (p=1/3)
GCRR-JR (p=1/2)
GCRR-JR (p=2/3)
0
Pricing Error
-0.05
-0.1
-0.15
-0.2
10
28
46
64
82
Number of Time Steps
This figure shows the convergence patterns of the JR model, GCRR-JR (p=1/2) model, GCRR-JR (p=1/3)
model and GCRR-JR (p=2/3) model. The asset price is 100, the strike price is 95, the volatility is 0.2, the
maturity of the option is six months, and the risk-free rate is 0.06.
8
100
Table 2 reports the prices of European call and American put options calculated
using the CRR model, the GCRR-XPC model, GCRR- X on the 1/3-th nodes model, and
GCRR- X on the 2/3-th nodes model The parameters are: the asset price is 40, the strike
price is 45, the asset price volatility is 0.4, the maturity of the option is six months, and
the risk-free rate is 0.6.
The errors of the CRR model do not necessarily go down as n increases. For
example, the error for the CRR model is 0.0001215 when n  6400 , worse than an error
of 0.0000966 when n  3200 . In contrast, the error decreases as n increases for the the
GCRR-XPC model, GCRR- X on the 1/3-th nodes model, and GCRR- X on the 2/3-th
nodes model. Thus, one can apply the Richardson extrapolation method to increase the
accuracy for these models.
Let e(n) be the pricing error of the n-step binomial model, i.e.
e(n)  Cn  CBS ,
(5)
where Cn is the n-step binomial price and C BS is the Black-Scholes price.4 Define the
error ratio as:
 (n)  e(n) e(2n).
(6)
The error ratio is a measure of the improvements in accuracy as the number of time steps
doubles. It is clear from Table 2 that the pricing error of the GCRR-XPC model is almost
exactly halved when the number of time steps doubles, i.e. the error ratio almost equals
two. In contrast, Table 2 indicates that the GCRR- X on the 1/3-th nodes model and the
GCRR- X on the 2/3-th nodes model have a slower convergence rate of order O (1 / n ) .
Similarly, in Table 3, the pricing error of the GCRR-JR (p=1/2) model is almost exactly
halved when the number of time steps doubles, i.e. the error ratio almost equals two. In
contrast, it seems that the GCRR-JR (p=1/3) model and the GCRR-JR (p=2/3) model
have a slower convergence rate of order O (1 / n ) . The extrapolation formulae when the
error ratio is known are suggested in the Appendix.
4
Since there is no exact solution for the American put, following Tian (1999) and Heston and Zhou (2000),
the pricing errors of the n-step binomial model is defined as: e(n)  Cn  Cn / 2 .
9
Table 2 Price and Error Ratios for European Calls and American Puts using the CRR Model and Three Related GCRR models
European Call
CRR
n Price
50
3.1127553
Error
GCRR-XPC
Error ratio
0.0197709
Price
Error
GCRR- X on the 1/3-th nodes
Error ratio Price
3.0761791 -0.0168054
Error
3.1172274 0.0242430
GCRR- X on the 2/3-th node
Error ratio Price
Error
Error ratio
3.0294775 -0.0635070
100
3.0872827 -0.0057017 -3.4675560
3.0845771 -0.0084073 1.9988942 3.1164068 0.0234224 1.0350322 3.0495034 -0.0434811 1.4605662
200
3.0891970 -0.0037874
1.5054227
3.0887797 -0.0042047 1.9994844 3.1109797 0.0179953 1.3015833 3.0650620 -0.0279225 1.5572070
400
3.0929439 -0.0000405 93.5404085
3.0908818 -0.0021026 1.9997516 3.1070312 0.0140468 1.2810991 3.0739472 -0.0190372 1.4667279
800
3.0925503 -0.0004341
0.0932659
3.0919330 -0.0010514 1.9998781 3.1033497 0.0103653 1.3551761 3.0801303 -0.0128541 1.4810200
1600
3.0934409
0.0004564 -0.9511233
3.0924587 -0.0005257 1.9999397 3.1006093 0.0076248 1.3594105 3.0841135 -0.0088710 1.4490152
3200
3.0930811
0.0000966
4.7226640
3.0927216 -0.0002629 1.9999699 3.0984943 0.0055099 1.3838448 3.0868519 -0.0061326 1.4465349
6400
B-S
price
3.0931059
0.0001215
0.7954713
3.0928530 -0.0001314 1.9999849 3.0969498 0.0039653 1.3895148 3.0887076 -0.0042768 1.4339211
3.0929844
3.0929844
3.0929844
3.0929844
American Put
n Price
50
7.0271842
100
7.0068793
200
7.0058851
Error
Error ratio
Price
Error
Error ratio Price
Error
Error ratio Price
Error
Error ratio
6.9997733
7.0165994
6.9792354
0.0203050
7.0036989 -0.0039255
7.0174403 -0.0008409
6.9880395 -0.0088041
0.0009942 20.4232027
7.0057155 -0.0020166 1.9465921 7.0155716 0.0018687 -0.4499987 6.9950865 -0.0070470 1.2493368
400
7.0084847 -0.0025996 -0.3824407
7.0067137 -0.0009982 2.0202401 7.0139161 0.0016555 1.1287822 6.9990891 -0.0040026 1.7606073
800
7.0076520
0.0008327 -3.1221175
7.0071955 -0.0004819 2.0714921 7.0123330 0.0015831 1.0457441 7.0018798 -0.0027906 1.4343098
1600
7.0082090 -0.0005569 -1.4950395
7.0074457 -0.0002501 1.9264892 7.0111153 0.0012178 1.2999743 7.0036826 -0.0018028 1.5479088
3200
7.0078375
0.0003715 -1.4993231
7.0075681 -0.0001224 2.0429200 7.0101693 0.0009460 1.2873403 7.0049213 -0.0012387 1.4553711
6400
7.0078186
0.0000189 19.6141612
7.0076292 -0.0000611 2.0050691 7.0094751 0.0006942 1.3626916 7.0057600 -0.0008387 1.4770350
Notes: The Parameters are: the asset price is 40, the strike price is 45, the asset price volatility is 0.4, the maturity of the option is six months, the risk-free rate is
0.6, and the number of time steps starts at 50 and doubles each time subsequently. This table reports price, pricing errors, and error ratios from the CRR model, the
GCRR-XPC model, the GCRR- X on the 1/3-th nodes model, the GCRR- X on the 2/3-th nodes model.
10
Table 3 Price and Error Ratios for European Calls and American Puts using the JR Model and Three Related GCRR models
n Price
50
100
200
400
800
1600
3200
6400
B-S
price
JR
Error
Price
3.1106130 0.0176285
3.0971533 0.0041689 4.2285686
3.0935241 0.0005397 7.7248534
3.0953566 0.0023722 0.2274975
3.0940317 0.0010473 2.2650576
3.0923781 -0.0006063 -1.7273667
3.0930670 0.0000826 -7.3408261
3.0928771 -0.0001073 -0.7696826
3.0646880
3.0808786
3.0867817
3.0896514
3.0914335
3.0921893
3.0925832
3.0927856
3.0929844
3.0929844
n Price
Error
50
100
7.0249121
7.0142375
0.0106746
200
7.0098612
0.0043763
400
800
1600
3200
6400
Error ratio
European Call
GCRR-JR (p=1/2)
Error
Error ratio Price
Error ratio
Price
6.9948508
7.0020738
2.4391967
7.0101279 -0.0002667 -16.4106629
7.0088470 0.0012808 -0.2082054
7.0074697 0.0013773 0.9299303
7.0078346 -0.0003649 -3.7744614
7.0076661 0.0001685 -2.1656175
-0.0282964
-0.0121058
-0.0062028
-0.0033330
-0.0015509
-0.0007951
-0.0004012
-0.0001988
2.3374246
1.9516768
1.8609907
2.1490994
1.9505708
1.9816991
2.0184492
GCRR-JR (p=1/3)
Error
Error ratio Price
3.1266496
3.1229406
3.1159382
3.1107778
3.1063754
3.1028496
3.1001488
3.0981502
0.0336651
0.0299562 1.1238120
0.0229538 1.3050670
0.0177934 1.2900141
0.0133909 1.3287666
0.0098652 1.3573885
0.0071644 1.3769746
0.0051658 1.3869016
3.0929844
Error
American Put
Error ratio Price
3.0002820
3.0342243
3.0554242
3.0676809
3.0759322
3.0812860
3.0849041
3.0873602
-0.09270245
-0.05876007 1.57764370
-0.03756026 1.56442129
-0.02530355 1.48438717
-0.01705221 1.48388688
-0.01169840 1.45765261
-0.00808034 1.44776122
-0.00562422 1.43670451
3.0929844
Error
-0.0072230
7.0162674
7.0171254
-0.0008580
7.0048184
-0.0027446 2.6316614
7.0154509
0.0016745
7.0061896
7.0069831
7.0073317
7.0075096
7.0076004
-0.0013712
-0.0007935
-0.0003486
-0.0001780
-0.0000908
7.0139527
7.0125108
7.0112810
7.0103178
7.0095929
2.0016892
1.7280424
2.2762930
1.9588599
1.9605003
GCRR-JR (p=2/3)
Error
Error ratio
Error ratio Price
0.5123652
0.0014982 1.1177210
0.0014419 1.0390038
0.0012298 1.1724385
0.0009631 1.2768974
0.0007249 1.3286790
Error
Error ratio
6.9709080
6.9846444 -0.01373646
6.9931022 -0.00845776 1.62412507
6.9979603
7.0011871
7.0032567
7.0046428
7.0055771
-0.00485807 1.74097083
-0.00322680 1.50553798
-0.00206967 1.55909214
-0.00138609 1.49316840
-0.00093430 1.48356412
Notes: The Parameters are: the asset price is 40, the strike price is 45, the asset price volatility is 0.4, the maturity of the option is six months, the risk-free rate is 0.6,
and the number of time steps starts at 50 and doubles each time subsequently. This table reports price, pricing errors, and error ratios from the JR model, the GCRR-JR
(p=1/3) model, the GCRR-JR (p=1/2) model, the GCRR-JR (p=2/3) model.
11
We also compare the numerical efficiency of the various GCRR models including
GCRR-XPC model, GCRR- X on the 1/3-th nodes model, GCRR- X on the 2/3-th nodes
model, GCRR-JR (p=1/2) model, GCRR-JR (p=1/3) model, and GCRR-JR (p=2/3)
model. The above binomial methods are appropriate for applying the Richardson
extrapolation to increase the accuracy.
All the above methods are applied to price a range of European options and
American options. The accuracy of each method is measured using the root mean squared
errors (RMSE) with respect to the closed-from solution, or the GCRR-XPC model with a
two-point extrapolation (n = 50000) in the case of American puts. The parameters of
these options are adopted from WAND (2002) as follows: the stock price is 40, the risk
free rate is 0.06, the strike prices are 35 and 45, the volatilities are 0.2 and 0.4, and the
maturities are 0.25 and 0.5.
To clarify the role played by Richardson extrapolation, we first compare the
numerical efficiency of each method on a raw basis without any extrapolation procedure.
The results are presented in Table 4. It is clear from Table 4 that GCRR-XPC outperforms
the other methods for pricing European calls and American puts and GCRR-JR (p=1/2)
model is the second best model. Moreover, the results of Table 4 also show that for
European call options, GCRR-XPC and GCRR-JR (p=1/2) model monotonically
converge to the Black-Scholes value at the 1 n rate. On the other hand, GCRR- X on the
1/3-th nodes model, GCRR- X on the 2/3-th nodes model, GCRR-JR (p=1/3) model and
GCRR-JR (p=2/3) model monotonically converge to the Black-Scholes value at the
1
n rate.
We compare the numerical efficiency of the various GCRR models when the
Richardson extrapolation technique is applied to all methods. The number of time steps
used for all models with a two-point extrapolation is (n, 2[n/4]) where [x] denotes the
Gaussian symbol. In Table 5, it is very clear that GCRR-XPC also outperforms the other
methods for pricing European calls and American puts and GCRR-JR (p=1/2) model is
the second best model when the Richardson extrapolation formula is applied to all
binomial models.
12
Table 4 A Comparison of the Performance of Various Models Over a Range of Option Types and
Parameters (without Extrapolation)
n
GCRR
GCRR- X on the GCRR- X on the
-XPC
1/3-th nodes
2/3-th nodes
GCRR-JR
GCRR-JR
GCRR-JR
(p=1/2)
(p=1/3)
(p=2/3)
Approximation RMS Error for European Calls
50
100
200
400
800
1600
3200
9.17E-03
2.55E-02
3.17E-02
4.59E-03
1.94E-02
2.26E-02
2.29E-03
1.34E-02
1.50E-02
1.15E-03
9.77E-03
1.06E-02
5.74E-04
6.91E-03
7.32E-03
2.87E-04
4.94E-03
5.15E-03
1.43E-04
3.51E-03
3.61E-03
6400
7.17E-05
2.49E-03
Approximation RMS Error for American Put
50
100
200
400
800
1600
3200
6400
2.54E-03
4.46E-03
1.70E-02
1.67E-02
2.24E-03
1.26E-02
1.25E-02
1.11E-03
8.51E-03
8.47E-03
5.50E-04
6.08E-03
6.06E-03
2.75E-04
4.25E-03
4.24E-03
1.37E-04
3.01E-03
3.00E-03
6.81E-05
2.12E-03
2.12E-03
3.39E-05
1.50E-03
1.50E-03
1.50E-02
2.18E-02
4.40E-02
7.05E-03
1.67E-02
2.82E-02
3.61E-03
1.22E-02
1.80E-02
1.85E-03
9.13E-03
1.21E-02
9.06E-04
6.72E-03
8.23E-03
4.56E-04
4.90E-03
5.64E-03
2.28E-04
3.53E-03
3.91E-03
1.14E-04
2.54E-03
2.72E-03
8.29E-03
2.12E-02
1.89E-02
3.84E-03
1.41E-02
1.30E-02
1.98E-03
9.36E-03
8.72E-03
9.76E-04
6.43E-03
6.15E-03
4.86E-04
4.46E-03
4.34E-03
2.45E-04
3.13E-03
3.06E-03
1.22E-04
2.20E-03
2.17E-03
6.07E-05
1.55E-03
1.53E-03
Note. The options test set for both European and American options is from WAND (2002) and consists of
combinations of strike prices of 35 and 45; maturities of 0.25 and 0.5, and volatilities of 0.2and 0.4. RMS errors
averaged over the eight different sets of parameters are reported. GCRR-XPC, GCRR- X on the 1/3-th nodes, GCRRX on the 2/3-th nodes, GCRR-JR (p=1/2), GCRR-JR (p=1/3), and GCRR-JR (p=2/3) models are as described in the
text.
13
Table 5 A Comparison of the Performance of Various Models Over a Range of Option Types and
Parameters (with Extrapolation)
n
GCRR
GCRR- X on the GCRR- X on the
-XPC
1/3-th nodes
2/3-th nodes
GCRR-JR
GCRR-JR
GCRR-JR
(p=1/2)
(p=1/3)
(p=2/3)
Approximation RMS Error for European Calls
50
7.61E-04
1.65E-02
2.05E-02
5.24E-03
2.52E-02
2.68E-02
100
2.95E-05
9.91E-03
7.73E-03
1.54E-03
1.23E-02
1.20E-02
200
7.47E-06
3.72E-03
4.68E-03
3.56E-04
5.14E-03
7.33E-03
400
1.88E-06
2.16E-03
1.85E-03
1.80E-04
3.12E-03
2.78E-03
800
4.72E-07
9.25E-04
1.05E-03
8.82E-05
1.52E-03
1.49E-03
1600
1.18E-07
5.08E-04
4.67E-04
2.15E-05
7.66E-04
7.63E-04
3200
2.96E-08
2.35E-04
2.51E-04
4.45E-06
3.75E-04
3.70E-04
6400
7.39E-09
1.24E-04
1.18E-04
2.83E-06
1.90E-04
1.84E-04
Approximation RMS Error for American Put
50
1.05E-03
1.14E-02
9.06E-03
3.83E-03
1.71E-02
1.36E-02
100
1.03E-04
5.43E-03
4.40E-03
1.09E-03
7.64E-03
6.59E-03
200
1.58E-04
2.48E-03
2.04E-03
4.38E-04
3.55E-03
3.28E-03
400
2.09E-05
1.09E-03
9.43E-04
1.28E-04
1.85E-03
1.62E-03
800
2.26E-05
5.44E-04
4.61E-04
4.56E-05
9.01E-04
7.81E-04
1600
5.94E-06
2.48E-04
2.27E-04
1.48E-05
4.23E-04
3.90E-04
3200
2.68E-06
1.25E-04
1.15E-04
3.67E-06
2.13E-04
1.99E-04
6400
7.21E-07
5.93E-05
5.71E-05
2.29E-06
1.02E-04
9.96E-05
Note. The options test set for both European and American options is from WAND (2002) and consists of
combinations of strike prices of 35 and 45; maturities of 0.25 and 0.5, and volatilities of 0.2and 0.4. RMS errors
averaged over the eight different sets of parameters are reported. GCRR-XPC, GCRR- X on the 1/3-th nodes, GCRRX on the 2/3-th nodes, GCRR-JR (p=1/2), GCRR-JR (p=1/3), and GCRR-JR (p=2/3) models are as described in the
text.
14
5. Conclusion
This paper provides a further analysis of the convergence rates and patterns based
on the GCRR model of Chung and Shih (2007). We discuss how to correct the
oscillatory behavior of the CRR model by various GCRR models so that the binomial
prices converge to its continuous time price monotonically and smoothly and thus one
can apply the Richardson extrapolation method to increase the accuracy for these models.
It is shown that when the risk-neutral probability p equals or converges to 0.5, then
the convergence rate is order of O (1/ n) . For example, GCRR-XPC and GCRR-JR
(p=1/2) model monotonically converge to the Black-Scholes value at the 1 n rate.
However, when p does note equal or converge to 0.5, the convergence rate is order of
O (1/ n ) . For example, GCRR- X on the 1/3-th nodes model, GCRR- X on the 2/3-th
nodes model, GCRR-JR (p=1/3) model, and GCRR-JR (p=2/3) model monotonically
converge to the Black-Scholes value at the 1/ n rate.
Finally, the numerical results indicate that the GCRR models with various
modifications are efficient for pricing a range of options. GCRR-XPC outperforms the
other GCRR models for pricing European calls and American puts and GCRR-JR (p=1/2)
model is the second best model.
15
Appendix: Extrapolation formulae for various GCRR models
Since the convergence pattern of the GCRR model is monotonic and smooth, we
apply the extrapolation technique to enhance its accuracy. We first establish a two-point
extrapolation formula using the GCRR binomial prices with geometrically increasing
time steps n and 2n. The Black-Scholes price can be exactly calculated as follows:
 (n)C2 n  Cn
(A1)
CBS 
,
 ( n)  1
where C2n and Cn are the binomial option prices according to the number of time steps
equal to 2n and n respectively. Although the error ratio  (n ) is generally unknown, we
can still apply the following extrapolation formula to improve accuracy as long as the
error ratio converges to a constant quickly enough: Cˆ 2n  ( C2n  Cn )/(   1) , where 
is the limit of the error ratio series. It is straightforward to prove that the pricing error
remaining in the extrapolated price follows:
   ( n)
(A2)
Cˆ 2 n C BS 
e(2n).
 1
   ( n)
In other words, the pricing error reduces from e(2n) to
e(2n) after a two-point
 1
extrapolation technique is applied.
16
References:
1. Broadie, M., and J. Detemple, 1996, American Option Valuation: New Bounds,
Approximations, and a Comparison of Existing Methods, Review of Financial Studies,
9, 1211-1250.
2. Chung, S. L., and P. T. Shih, 2007, Generalized Cox-Ross-Rubinstein Binomial
Models, Management Science, 53, 508-520.
3. Cox, J. C., S. A. Ross, and M. Rubinstein, 1979, Option Pricing: a Simplified
Approach, Journal of Financial Economics, 7, 229-263.
4. Diener, F., and M. Diener, 2004, Asymptotics of the Price Oscillations of a European
Call Option in a Tree Model, Mathematical Finance, 14, 271-293.
5. Figlewski, S. and B. Gao, 1999, The Adaptive Mesh Model: a New Approach to
Efficient Option Pricing, Journal of Financial Economics, 53, 313-351.
6. Heston, S., and G. Zhou, 2000, On the Rate of Convergence of Discrete-Time
Contingent Claims, Mathematical Finance, 10, 53-75.
7. Jarrow, R., and A. Rudd, 1983, Option Pricing, Homewood, IL: Richard D. Irwin.
8. Leisen, D. P. J., and M. Reimer, 1996, Binomial Models for Option Valuation –
Examining and Improving Convergence, Applied Mathematical Finance, 3, 319-346.
9. Omberg, E., 1988, Efficient Discrete Time Jump Process Models in Option Pricing,
Journal of Financial Quantitative Analysis, 23, 161-174.
10. Rendleman, R. J. and B. J. Bartter, 1979, Two-State Option Pricing, Journal of
Finance, 34, 1093-1110.
11. Ritchken, P., 1995, On Pricing Barrier Options, The Journal of Derivatives, 3, 19-28.
12. Tian, Y., 1993, A Modified Lattice Approach to Option Pricing, Journal of Futures
Markets, 13, 563-577.
13. Tian, Y., 1999, A Flexible Binomial Option Pricing Model, Journal of Futures
Markets, 19, 817-843.
14. Widdicks, M., A. D. Andricopoulos, D. P. Newton, and P. W. Duck, 2002, On the
Enhanced Convergence of Standard Lattice Methods for Option Pricing, Journal of
Futures Markets, 22, 315-338.
17